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Removing $v$ from $T$ exposes the $4$-cycle $abcd$, which we +retriangulate by adding one of the two diagonals $(a, c)$ or $(b, d)$. +We call this operation \emph{contraction at $v$ along diagonal +$(a, c)$}, denoted $T_{v, (a, c)}$. The contraction is \emph{valid} when +the chosen diagonal is not already an edge of $T$. + +\begin{lemma}[Good contraction] +\label{lem:good-contraction} +Let $T$ be a plane triangulation on $n \geq 7$ vertices with minimum +degree at least $4$. Then there exist a degree-$4$ vertex $v \in V(T)$, +with cyclic neighbors $a, b, c, d$, and an unordered pair +$\{a, c\}$ such that: +\begin{enumerate} +\item $(a, c) \not\in E(T)$; +\item $\deg_T(b) \geq 5$ and $\deg_T(d) \geq 5$. +\end{enumerate} +Under these conditions $T_{v, (a, c)}$ is a plane triangulation on +$n - 1$ vertices with minimum degree at least $4$. +\end{lemma} + +The conditions of Lemma~\ref{lem:good-contraction} ensure that the +contraction is valid (1) and md$_4$-preserving (2): the only vertices +whose degree changes under $T \to T_{v, (a, c)}$ are $a, b, c, d$, with +$\deg(a)$ and $\deg(c)$ unchanged (each loses the edge to $v$ but gains +the edge from the diagonal), while $\deg(b)$ and $\deg(d)$ each decrease +by $1$. + +The lemma is empirically true at $n = 7, \ldots, 11$ for every md$_4$ +iso-class; we conjecture it holds for all $n \geq 7$. The $n = 6$ case +is excluded: the unique md$_4$ iso-class is the octahedron, in which +every vertex has all four cyclic neighbors at degree $4$ and so no +contraction preserves md$_4$. The octahedron is therefore the base case +of the proposed induction. + +\begin{lemma}[Lift] +\label{lem:lift} +Let $T$ be a plane triangulation with minimum degree at least $4$, and +suppose Lemma~\ref{lem:good-contraction} applies via vertex $v$ and +diagonal $(a, c)$ with $T_{v, (a, c)}$ the resulting contraction. Let +$H$ be a plane triangulation on $V(T_{v, (a, c)}) = V(T) \setminus \{v\}$ +and $S$ a level source of $H$ such that the algorithm of +Section~\ref{sec:flip-algorithm} applied to $(H, S)$ produces +$T_{v, (a, c)}$ as a labelled simple graph. Define the \emph{lift} +$G \;:=\; H[a, b, c, d, v]$ by: +\begin{itemize} +\item adding vertex $v$ to $V(H)$; +\item removing the edge $(a, c)$ from $E(H)$; +\item adding the four edges $(v, a), (v, b), (v, c), (v, d)$. +\end{itemize} +Then $G$ is a plane triangulation on $|V(T)|$ vertices, and the +algorithm of Section~\ref{sec:flip-algorithm} applied to $(G, S)$ +produces $T$. +\end{lemma} + +Lemma~\ref{lem:lift} requires that $(a, c) \in E(H)$ and that the two +triangles of $H$ bordering $(a, c)$ have boundary +$\{a, b, c, d\}$. When these conditions hold, the lift restores a +degree-$4$ vertex $v$ inserted into the quadrilateral $abcd$; when they +fail, the lift is undefined and a different labelled preimage $H$ must +be chosen. + +\paragraph{Inductive scheme.} +Conjecture~\ref{conj:simple-md4} would follow from +Lemmas~\ref{lem:good-contraction} and~\ref{lem:lift} together with the +existence at each step of a labelled preimage $H$ satisfying the lift's +side conditions. The base case is the octahedron at $n = 6$, which is +empirically a simple level resolution +(Observation~\ref{obs:md4-simple-resolution}). The inductive step takes +an md$_4$ target $T$ on $n$ vertices, applies +Lemma~\ref{lem:good-contraction} to obtain an md$_4$ contraction +$T_{v, (a, c)}$ on $n - 1$ vertices, invokes the inductive hypothesis to +produce a labelled preimage $H$, and applies Lemma~\ref{lem:lift} to +lift $H$ to $G$ with $\mathrm{alg}(G, S) = T$. + +We have verified the entire scheme by hand for the unique md$_4$ +iso-class at $n = 7$: contraction at $v = 2$ along diagonal $(4, 3)$ +yields the octahedron on six vertices labelled $\{0, 1, 3, 4, 5, 6\}$; +a labelled preimage $H$ exists with source $S = \{0, 1, 6\}$; lifting +along $(4, 3, v = 2)$ produces a triangulation $G$ on seven vertices on +which the algorithm with source $S$ recovers $T$ exactly. The principal +remaining work is a proof of Lemma~\ref{lem:good-contraction} for all +$n \geq 7$, a proof of Lemma~\ref{lem:lift} (which involves analysing +how the algorithm's depth-guided flips interact with the added vertex +$v$), and a guarantee that a label-faithful preimage $H$ always +exists. + \begin{question} \label{q:terminate-all-n} Does Phase~1 terminate for all $(G, S)$? Equivalently, is there an